Positivity 4: 205–212, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
A Characterization of Operators Preserving
Disjointness in Terms of their Inverse
and A. KITOVER
Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202, USA
Department of Mathematics, CCP, Philadelphia, PA 19130, USA
Abstract. The characterization mentioned in the title is found.
Recall that a (linear) operator T : X → Y between vector lattices is disjoint-
ness preserving if T sends elements disjoint in X to elements disjoint in Y .If
T is a bijective disjointness preserving operator between Banach lattices, then a
well known theorem by Huijsmans–de Pagter  and Koldunov  asserts that
the inverse T
: Y → X is also disjointness preserving. Many other results
describing various conditions under which T
is disjointness preserving can be
found in . It was believed for a while that the same conclusion should remain
true for disjointness preserving operators between arbitrary vector lattices, or at
least, for operators between Dedekind complete vector lattices. However, as has
been recently shown by the authors [4,5], this is not true in general. This means, in
particular, that if one wants to ﬁnd a characterization of a disjointness preserving
operator in terms of its inverse, then a different condition is needed rather than
It is the purpose of this note to present such a condition. The authors would like
to express their thanks to Beata Randrianantoanina for her help in identifying this
condition. In her talk
devoted to description of non-surjective isometries between
some Orlicz spaces and based on her work , Randrianantoanina introduced an
interesting monotonicity condition and asked if it implied disjointness preservation.
The essence of this condition is as follows: if the support of a measurable function
is contained in the support of another measurable function x
, then the same
is true for the supports of their images, that is, the support of Tx
in the support of Tx
,whereT is the isometry in question. An abstract order-
theoretic version of this condition will be introduced in Deﬁnition 2.2 and denoted
by (β). As Examples 2.5 and 2.6 demonstrate, condition (β) and disjointness pre-
servation are independent in general. Nevertheless, these conditions are related
Delivered at the conference Function Spaces, held in Edwardsville in May of 1998.